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Mathematics in Everyday Life

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Mathematics in Everyday Life Gilad Lerman Department of Mathematics University of Minnesota Highland park elementary (6th graders) What do mathematicians do? – PowerPoint PPT presentation

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Title: Mathematics in Everyday Life


1
Mathematics in Everyday Life
  • Gilad Lerman
  • Department of Mathematics
  • University of Minnesota

Highland park elementary (6th graders)
2
What do mathematicians do?
What homework do I give my students?
  • Example of a recent homework Denoising

3
What do mathematicians do?
What projects do I assign my students?
  • Example of a recent project
  • Recognizing Panoramas
  • Panorama
  • How to obtain a panorama?

wide view of a physical space
4
How to obtain a panorama
  • By rotating line camera
  • Stitching together multiple images
  • Your camera can do it this way
  • E.g. PhotoStitch (Canon PowerShot SD600)

5
Experiment with PhotoStitch
Input 10 images along a bridge
Experiment done by Rebecca Szarkowski
6
Experiment continued
Output Panorama (PhotoStitch)
Output Panorama (by a more careful mathematical
algorithm)
Experiment done by Rebecca Szarkowski
7
Whats math got to do with it?
New Topic Relation of Imaging and Mathematics
From visual images to numbers (or digital images)
8
Digital Image Acquisition
9
From Numbers to Images
  • Let us type the following numbers
  • 1 1 1 1 1 1 1 1
  • 2 2 2 2 2 2 2 2
  • 3 3 3 3 3 3 3 3
  • 4 4 4 4 4 4 4 4
  • 5 5 5 5 5 5 5 5
  • 6 6 6 6 6 6 6 6
  • 7 7 7 7 7 7 7 7
  • 8 8 8 8 8 8 8 8
  • We then color them so 1black, 8white
  • rest of colors are in between

10
One more time
  • Now well try the following numbers
  • 1 1 1 1 1 1 1 1
  • 2 2 2 2 2 2 2 2
  • 4 4 4 4 4 4 4 4
  • 8 8 8 8 8 8 8 8
  • 16 16 16 16 16 16 16 16
  • 32 32 32 32 32 32 32 32
  • 64 64 64 64 64 64 64 64
  • 128 128 128 128 128 128 128 128
  • We then color them so 1black, 128white
  • rest of colors are in between

11
Lets compare
  • 1 1 1 1 1 1 1 1
  • 2 2 2 2 2 2 2 2
  • 3 3 3 3 3 3 3 3
  • 4 4 4 4 4 4 4 4
  • 5 5 5 5 5 5 5 5
  • 6 6 6 6 6 6 6 6
  • 7 7 7 7 7 7 7 7
  • 8 8 8 8 8 8 8 8

1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
4 4 4 4 4 4 4 4
8 8 8 8 8 8 8 8
16 16 16 16 16 16 16 16 32 32
32 32 32 32 32 32 64 64 64 64
64 64 64 64 128 128 128 128 128 128 128
128
12
From an Image to Its Numbers
  • We start with clown image
  • It has 200320 numbers
  • I cant show you all
  • Lets zoom on eye (4050)

13
Image to Numbers (Continued)
  • Well zoom on middle of eye image (1010)

14
The Numbers (Continued)
  • The middle of eye image (1010)

80 81 80 80 80 80 77 77
37 11 81 80 81 80 80 80
77 37 9 6 80 80 80 80
80 80 37 11 2 11 80 80
80 80 80 77 66 66 66 54
80 80 80 80 77 77 77 80
77 80 80 80 79 77 66 54
66 77 66 54 77 80 77 70
22 57 51 70 51 70 77 73
70 22 2 2 22 37 37 22
77 77 54 37 1 6 2 8
2 6 77 70 70 22 2 2
6 8 8 6 Note the rule
Bright colors high numbers Dark colors -
low numbers
15
More Relation of Imaging and Math
  • Averaging numbers ? smoothing images
  • Idea of averaging
  • take an image
  • Replace each point by
  • average with its neighbors
  • For example, 2 has the neighborhood
  • So replace 2 by

80 81 80 80 80 80 77 77
37 11 81 80 81 80 80 80
77 37 9 6 80 80 80 80
80 80 37 11 2 11 80 80
80 80 80 77 66 66 66 54
80 80 80 80 77 77 77 80
77 80 80 80 79 77 66 54
66 77 66 54 77 80 77 70
22 57 51 70 51 70 77 73
70 22 2 2 22 37 37 22
77 77 54 37 1 6 2 8
2 6 77 70 70 22 2 2
6 8 8 6
80 81 80 80 80 80 77 77
37 11 81 80 81 80 80 80
77 37 9 6 80 80 80 80
80 80 37 11 2 11 80 80
80 80 80 77 66 66 66 54
80 80 80 80 77 77 77 80
77 80 80 80 79 77 66 54
66 77 66 54 77 80 77 70
22 57 51 70 51 70 77 73
70 22 2 2 22 37 37 22
77 77 54 37 1 6 2 8
2 6 77 70 70 22 2 2
6 8 8 6
70 22 57 22 2 2 37 1
6
16
Example Smoothing by averaging
Original image on top left It is then averaged
with neighbors of distances 3, 5, 19, 15, 35, 45
17
Example Smoothing by averaging
And removing wrinkles by both.
18
More Relation of Imaging and Math
  • Differences of numbers ? sharpening images

On left image of moon On right its edges
(obtained by differences) We can add the two to
get a sharpened version of the first
19
Moon sharpening (continued)
20
Real Life Applications
  • Many
  • From a Minnesota based company
  • Their main job maintaining railroads
  • Main concern Identify cracks in railroads,
  • before too late

21
How to detect damaged rails?
  • Traditionally drive along the rail (very long)
    and inspect
  • Very easy to miss defects (falling asleep)
  • New technology getting pictures of rails

22
Millions of images then collected
23
How to detect Cracks?
  • Human observation
  • Train a computer
  • Recall that differences detect edges

Work done by Kyle Heuton (high school student at
Saint Paul)
24
Summary
  • Math is useful (beyond the grocery store)
  • Images are composed of numbers
  • Good math ideas ? good image processing
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